Ordinary bitmaps allow pixels to be black or white. The authors introduce a second bitmap, the 'alpha' bitmap, which allows pixels to be transparent as well. The alpha bitmap makes it possible to have black-and-white images that are nonrectangular or that have holes in them. It also provides a richer set of operations for working with bitmaps. The article presents the mathematics for a two-bit compositing algebra, and suggest extensions for two-bit compositing, painting, and region filling. Each of these operations can be implemented with ordinary bitbits and presented on ordinary bitmap displays. The authors analyze the cost of each two-bit operation in terms of the number of bitbits it requires